NavList:

Hi,

I would like to propose a yet simpler procedure for testing the scales intended for a Bygrave.

For a demonstration you might want to pull up Gary LaPook's Cotangent scale for his Flat Bygrave (107501.f2-lapook1).

In this case there are 38 horizontal lines. Let's give each line an id nr., for instance n=0 for the bottom line until n=37 for the top line.

Let me draw with a pointed pencil a straight line that crosses all the printed lines. So, in each line there is now a (single) cross point of each of those prior printed lines (n=0 to 37)  of Gary's scale and my pencil line. 

I now proceed to predict the angles at which the cross points occur. If the predicted cross points concur with the predicted ones, we have a reason for the assumption the entire Cotangent scale is probably OK. This is particularly so if I choose for the pencil line a line close to a diagonal since it would be affected by horizontal and by vertical errors of Gary's printer  - if any.

For convenience I choose the angles 89 deg on line 36 and 1 deg. on line 0 and connect them with my pencil line. It passes through integer angle values at each end and is close to the top-left to right bottom diagonal.

I don't want to bore you with the math. The final formulae are as follows: 

A.)  Define r = log( cotg(1)/cotg(89) ) / 36;         %  angles: in deg; log: to basis e

B.)  Then C.P. = acotg ( cotg(1)*exp(-r*n) );      % C.P. = angles at the cross points for n=0 to 36 

And here are the results. The cross points should be located at:

Now to the Bygrave: Here, the pencil line needs to be parallel to the axis of the drums - no other change is necessary. Any tilted line on the flat scale would otherwise become a spiral on a drum - difficult to make.

If there is interest, I will show you the results for the Cosinus scale as well, and I will give you the math behind he formulas.

I hope this note will be found useful

H